Link to actual problem [1261] \[ \boxed {\left (3 x +2\right ) y^{\prime \prime }-y^{\prime } x +2 y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}
With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x \left (i \sqrt {23}-1\right )}{6}} \left (3 x +2\right )^{\frac {7}{9}} \operatorname {KummerM}\left (\frac {8}{9}-\frac {11 i \sqrt {23}}{207}, \frac {16}{9}, \frac {i \sqrt {23}\, \left (3 x +2\right )}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (i \sqrt {23}-1\right )}{6}} y}{\left (3 x +2\right )^{\frac {7}{9}} \operatorname {KummerM}\left (\frac {8}{9}-\frac {11 i \sqrt {23}}{207}, \frac {16}{9}, \frac {i \sqrt {23}\, \left (3 x +2\right )}{9}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x \left (i \sqrt {23}-1\right )}{6}} \left (3 x +2\right )^{\frac {7}{9}} \operatorname {KummerU}\left (\frac {8}{9}-\frac {11 i \sqrt {23}}{207}, \frac {16}{9}, \frac {i \sqrt {23}\, \left (3 x +2\right )}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (i \sqrt {23}-1\right )}{6}} y}{\left (3 x +2\right )^{\frac {7}{9}} \operatorname {KummerU}\left (\frac {8}{9}-\frac {11 i \sqrt {23}}{207}, \frac {16}{9}, \frac {i \sqrt {23}\, \left (3 x +2\right )}{9}\right )}\right ] \\ \end{align*}